DATA MINING LECTURE 6 Sketching, Min-Hashing, Locality Sensitive Hashing

MIN-HASHING AND LOCALITY SENSITIVE HASHING Thanks to: Rajaraman, Ullman, Lekovec “Mining Massive Datasets” Evimaria Terzi, slides for Data Mining Course.

Motivating problem Find duplicate and near-duplicate documents from a web crawl. If we wanted exact duplicates we could do this by hashing We will see how to adapt this technique for near duplicate documents

Main issues What is the right representation of the document when we check for similarity? E.g., representing a document as a set of characters will not do (why?) When we have billions of documents, keeping the full text in memory is not an option. We need to find a shorter representation How do we do pairwise comparisons of billions of documents? If exact match was the issue it would be ok, can we replicate this idea?

5 The Big Picture Shingling Docu- ment The set of strings of length k that appear in the doc- ument Minhash- ing Signatures : short integer vectors that represent the sets, and reflect their similarity Locality- sensitive Hashing Candidate pairs : those pairs of signatures that we need to test for similarity.

Shingling Shingle: a sequence of k contiguous characters a rose is rose is a ose is a r se is a ro e is a ros is a rose s a rose i a rose is Set of ShinglesSet of 64-bit integers Hash function (Rabin’s fingerprints)

7 Basic Data Model: Sets Document: A document is represented as a set of shingles (more accurately, hashes of shingles) Document similarity: Jaccard similarity of the sets of shingles. Common shingles over the union of shingles Sim (C 1, C 2 ) = |C 1  C 2 |/|C 1  C 2 |. Applicable to any kind of sets. E.g., similar customers or items.

Signatures Key idea: “hash” each set S to a small signature Sig (S), such that: 1. Sig (S) is small enough that we can fit all signatures in main memory. 2. Sim (S 1, S 2 ) is (almost) the same as the “similarity” of Sig (S 1 ) and Sig (S 2 ). (signature preserves similarity). Warning: This method can produce false negatives, and false positives (if an additional check is not made). False negatives: Similar items deemed as non-similar False positives: Non-similar items deemed as similar

9 From Sets to Boolean Matrices Represent the data as a boolean matrix M Rows = the universe of all possible set elements In our case, shingle fingerprints take values in [0… ] Columns = the sets In our case, documents, sets of shingle fingerprints M(r,S) = 1 in row r and column S if and only if r is a member of S. Typical matrix is sparse. We do not really materialize the matrix

Example XY A11 B10 C00 D00 E01 F11 G11

XY A11 B10 C00 D00 E01 F11 G11 At least one of the columns has value 1

Example XY A11 B10 C00 D00 E01 F11 G11 Both columns have value 1

13 Minhashing Pick a random permutation of the rows (the universe U). Define “hash” function for set S h(S) = the index of the first row (in the permuted order) in which column S has 1. or equivalently h(S) = the index of the first element of S in the permuted order. Use k (e.g., k = 100) independent random permutations to create a signature.

Example of minhash signatures Input matrix S1S1 S2S2 S3S3 S4S4 A1010 B1001 C0101 D0101 E0111 F1010 G1010 A C G F B E D S1S1 S2S2 S3S3 S4S4 1A1010 2C0101 3G1010 4F1010 5B1001 6E0111 7D Random Permutation

Example of minhash signatures Input matrix S1S1 S2S2 S3S3 S4S4 A1010 B1001 C0101 D0101 E0111 F1010 G1010 D B A C F G E S1S1 S2S2 S3S3 S4S4 1D0101 2B1001 3A1010 4C0101 5F1010 6G1010 7E Random Permutation

Example of minhash signatures Input matrix S1S1 S2S2 S3S3 S4S4 A1010 B1001 C0101 D0101 E0111 F1010 G1010 C D G F A B E S1S1 S2S2 S3S3 S4S4 1C0101 2D0101 3G1010 4F1010 5A1010 6B1001 7E Random Permutation

Example of minhash signatures Input matrix S1S1 S2S2 S3S3 S4S4 A1010 B1001 C0101 D0101 E0101 F1010 G1010 S1S1 S2S2 S3S3 S4S4 h1h h2h h3h ≈ Sig(S) = vector of hash values e.g., Sig(S 2 ) = [2,1,1] Sig(S,i) = value of the i-th hash function for set S E.g., Sig(S 2,3) = 1 Signature matrix

18 Hash function Property Pr(h(S 1 ) = h(S 2 )) = Sim(S 1,S 2 ) where the probability is over all choices of permutations. Why? The first row where one of the two sets has value 1 belongs to the union. Recall that union contains rows with at least one 1. We have equality if both sets have value 1, and this row belongs to the intersection

Example Universe: U = {A,B,C,D,E,F,G} X = {A,B,F,G} Y = {A,E,F,G} Union = {A,B,E,F,G} Intersection = {A,F,G} XY A11 B10 C00 D00 E01 F11 G11 D * * C * * * XY D00 C00 Rows C,D could be anywhere they do not affect the probability

Example Universe: U = {A,B,C,D,E,F,G} X = {A,B,F,G} Y = {A,E,F,G} Union = {A,B,E,F,G} Intersection = {A,F,G} XY A11 B10 C00 D00 E01 F11 G11 D * * C * * * XY D00 C00 The * rows belong to the union

Example Universe: U = {A,B,C,D,E,F,G} X = {A,B,F,G} Y = {A,E,F,G} Union = {A,B,E,F,G} Intersection = {A,F,G} XY A11 B10 C00 D00 E01 F11 G11 D * * C * * * XY D00 C00 The question is what is the value of the first * element

Example Universe: U = {A,B,C,D,E,F,G} X = {A,B,F,G} Y = {A,E,F,G} Union = {A,B,E,F,G} Intersection = {A,F,G} XY A11 B10 C00 D00 E01 F11 G11 D * * C * * * XY D00 C00 If it belongs to the intersection then h(X) = h(Y)

Example Universe: U = {A,B,C,D,E,F,G} X = {A,B,F,G} Y = {A,E,F,G} Union = {A,B,E,F,G} Intersection = {A,F,G} XY A11 B10 C00 D00 E01 F11 G11 D * * C * * * XY D00 C00

Zero similarity is preserved High similarity is well approximated 24 Similarity for Signatures The similarity of signatures is the fraction of the hash functions in which they agree. With multiple signatures we get a good approximation S1S1 S2S2 S3S3 S4S4 A1010 B1001 C0101 D0101 E0101 F1010 G1010 S1S1 S2S2 S3S3 S4S ≈ ActualSig (S1, S2)(S1, S2)00 (S 1, S 3 )3/52/3 (S 1, S 4 )1/70 (S 2, S 3 )00 (S 2, S 4 )3/41 (S 3, S 4 )00 Signature matrix

Is it now feasible? Assume a billion rows Hard to pick a random permutation of 1…billion Even representing a random permutation requires 1 billion entries!!! How about accessing rows in permuted order? 

Being more practical Instead of permuting the rows we will apply a hash function that maps the rows to a new (possibly larger) space The value of the hash function is the position of the row in the new order (permutation). Each set is represented by the smallest hash value among the elements in the set The space of the hash functions should be such that if we select one at random each element (row) has equal probability to have the smallest value Min-wise independent hash functions

Algorithm – One set, one hash function Computing Sig(S,i) for a single column S and single hash function h i for each row r compute h i (r ) if column S that has 1 in row r if h i (r ) is a smaller value than Sig(S,i) then Sig(S,i) = h i (r); Sig(S,i) will become the smallest value of h i (r) among all rows (shingles) for which column S has value 1 (shingle belongs in S); i.e., h i (r) gives the min index for the i-th permutation In practice only the rows (shingles) that appear in the data h i (r) = index of row r in permutation S contains row r Find the row r with minimum index

Algorithm – All sets, k hash functions Pick k=100 hash functions (h 1,…,h k ) for each row r for each hash function h i compute h i (r ) for each column S that has 1 in row r if h i (r ) is a smaller value than Sig(S,i) then Sig(S,i) = h i (r); In practice this means selecting the hash function parameters Compute h i (r) only once for all sets

29 Example RowS1S2 A 1 0 B 0 1 C 1 1 D 1 0 E 0 1 h(x) = x+1 mod 5 g(x) = 2x+3 mod 5 h(0) = 11- g(0) = 33- h(1) = 212 g(1) = 030 h(2) = 312 g(2) = 220 h(3) = 412 g(3) = 420 h(4) = 010 g(4) = 120 Sig1Sig2 Row S1S2 E 0 1 A 1 0 B 0 1 C 1 1 D 1 0 Row S1S2 B 0 1 E 0 1 C 1 0 A 1 1 D 1 0 x01234x01234 h(Row) g(Row) h(x) g(x)

30 Implementation Often, data is given by column, not row. E.g., columns = documents, rows = shingles. If so, sort matrix once so it is by row. And always compute h i (r ) only once for each row.

31 Finding similar pairs Problem: Find all pairs of documents with similarity at least t = 0.8 While the signatures of all columns may fit in main memory, comparing the signatures of all pairs of columns is quadratic in the number of columns. Example: 10 6 columns implies 5*10 11 column- comparisons. At 1 microsecond/comparison: 6 days.

32 Locality-Sensitive Hashing What we want: a function f(X,Y) that tells whether or not X and Y is a candidate pair: a pair of elements whose similarity must be evaluated. A simple idea: X and Y are a candidate pair if they have the same min-hash signature. Easy to test by hashing the signatures. Similar sets are more likely to have the same signature. Likely to produce many false negatives. Requiring full match of signature is strict, some similar sets will be lost. Improvement: Compute multiple signatures; candidate pairs should have at least one common signature. Reduce the probability for false negatives. ! Multiple levels of Hashing!

33 Signature matrix reminder Matrix M n hash functions Sig(S): signature for set S hash function i Sig(S,i) signature for set S’ Sig(S ’,i) Prob(Sig(S,i) == Sig(S’,i)) = sim(S,S’)

34 Partition into Bands – (1) Divide the signature matrix Sig into b bands of r rows. Each band is a mini-signature with r hash functions.

35 Partitioning into bands Matrix Sig r rows per band b bands One signature n = b*r hash functions b mini-signatures

36 Partition into Bands – (2) Divide the signature matrix Sig into b bands of r rows. Each band is a mini-signature with r hash functions. For each band, hash the mini-signature to a hash table with k buckets. Make k as large as possible so that mini-signatures that hash to the same bucket are almost certainly identical.

37 Matrix M r rows b bands Hash Table Columns 2 and 6 are (almost certainly) identical. Columns 6 and 7 are surely different.

38 Partition into Bands – (3) Divide the signature matrix Sig into b bands of r rows. Each band is a mini-signature with r hash functions. For each band, hash the mini-signature to a hash table with k buckets. Make k as large as possible so that mini-signatures that hash to the same bucket are almost certainly identical. Candidate column pairs are those that hash to the same bucket for at least 1 band. Tune b and r to catch most similar pairs, but few non- similar pairs.

39 Analysis of LSH – What We Want True similarity s of two sets Probability of sharing a bucket t No chance if s < t Probability = 1 if s > t

40 What One Band of One Row Gives You Similarity s of two sets Probability of sharing a bucket t Remember: probability of equal hash-values = similarity Single hash signature Prob(Sig(S,i) == Sig(S’,i)) = sim(S,S’)

41 What b Bands of r Rows Gives You Similarity s of two sets Probability of sharing a bucket t s rs r All rows of a band are equal 1 - Some row of a band unequal ()b)b No bands identical 1 - At least one band identical t ~ (1/b) 1/r

42 Example: b = 20; r = 5 s 1-(1-s r ) b t = 0.5

43 Suppose S 1, S 2 are 80% Similar We want all 80%-similar pairs. Choose 20 bands of 5 integers/band. Probability S 1, S 2 identical in one particular band: (0.8) 5 = Probability S 1, S 2 are not similar in any of the 20 bands: ( ) 20 = i.e., about 1/3000-th of the 80%-similar column pairs are false negatives. Probability S 1, S 2 are similar in at least one of the 20 bands: = 0.999

44 Suppose S 1, S 2 Only 40% Similar Probability S 1, S 2 identical in any one particular band: (0.4) 5 = Probability S 1, S 2 identical in at least 1 of 20 bands: ≤ 20 * 0.01 = 0.2. But false positives much lower for similarities << 40%.

45 LSH Summary Tune to get almost all pairs with similar signatures, but eliminate most pairs that do not have similar signatures. Check in main memory that candidate pairs really do have similar signatures. Optional: In another pass through data, check that the remaining candidate pairs really represent similar sets.

Locality-sensitive hashing (LSH) Big Picture: Construct hash functions h: R d  U such that for any pair of points p,q, for distance function D we have: If D(p,q)≤r, then Pr[h(p)=h(q)] ≥ α is high If D(p,q)≥cr, then Pr[h(p)=h(q)] ≤ β is small Then, we can find close pairs by hashing LSH is a general framework: for a given distance function D we need to find the right h h is (r,cr, α, β)-sensitive

47 LSH for Cosine Distance For cosine distance, there is a technique analogous to minhashing for generating a ( d 1,d 2,(1-d 1 /180),(1-d 2 /180) ) - sensitive family for any d 1 and d 2. Called random hyperplanes.

48 Random Hyperplanes Pick a random vector v, which determines a hash function h v with two buckets. h v (x) = +1 if v.x > 0; = -1 if v.x < 0. LS-family H = set of all functions derived from any vector. Claim: Prob[h(x)=h(y)] = 1 – (angle between x and y)/180

49 Proof of Claim x y Look in the plane of x and y. θ h v (x) = +1 h v (x) = -1 For a random vector v the values of the hash functions h v (x) and h v (y) depend on where the vector v falls h v (y) = -1 h v (y) = +1 h v (x) ≠ h v (y) when v falls into the shaded area. What is the probability of this for a randomly chosen vector v? θ θ P[h v (x) ≠ h v (y)] = 2θ/360 = θ/180 P[h v (x) = h v (y)] = 1- θ/180

50 Signatures for Cosine Distance Pick some number of vectors, and hash your data for each vector. The result is a signature (sketch ) of +1’s and – 1’s that can be used for LSH like the minhash signatures for Jaccard distance.

51 Simplification We need not pick from among all possible vectors v to form a component of a sketch. It suffices to consider only vectors v consisting of +1 and –1 components.